# What is the difference between a linear and non-linear solution in the bending of beams?

I have been working on a simulator for bending of beams and came now to a tricky doubt: What should be the difference between a linear and non linear solution in this case (graphic at bottom)?

The solution of the following ODE gives us the non-linear curvature:

and for very small angles (dy/dx)^2 will tend to zero, so we can linearize (Sorry, $dy=dv$ in this image):

So we integrate the following equation (I used the bvp4c function on Matlab) , that includes the curvature, to obtain the deflection of the beam:

In red is the non-linear solution and in Blue the linear solution.

My doubt is: at the middle of the $x$-axis $dy/dx=0$ in both curves, should I expect then also to have the same value of $y$, since at that precise point I could also cancel $dy/dx$ in the formula and both equations would look the same? In other words, what should I expect as a difference between a linear and a non-linear solution in such equations?

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